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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X3,10,4,11 X15,5,16,20 X7,17,8,16 X17,12,18,13 X9,14,10,15 X13,18,14,19 X19,9,20,8 X2536 X11,4,12,1

Gauss Code:

{{1, -9, -2, 10}, {9, -1, -4, 8, -6, 2, -10, 5, -7, 6, -3, 4, -5, 7, -8, 3}}

Jones Polynomial:

2q-13/2 - 5q-11/2 + 6q-9/2 - 8q-7/2 + 7q-5/2 - 7q-3/2 + 5q-1/2 - 3q1/2 + q3/2

A2 (sl(3)) Invariant:

- q-24 - 2q-20 + q-18 + q-16 + 2q-14 + 5q-12 + q-10 + 4q-8 - q-6 - 1 + q2 - q4

HOMFLY-PT Polynomial:

az-1 + 2az + 3az3 + az5 - 4a3z-1 - 9a3z - 9a3z3 - 5a3z5 - a3z7 + 4a5z-1 + 6a5z + 4a5z3 + a5z5 - a7z-1 - a7z

Kauffman Polynomial:

1 - 3z2 + 3z4 - z6 - az-1 + 3az - 9az3 + 10az5 - 3az7 + 4a2 - 14a2z2 + 13a2z4 + a2z6 - 2a2z8 - 4a3z-1 + 15a3z - 28a3z3 + 29a3z5 - 9a3z7 + 7a4 - 21a4z2 + 21a4z4 - 3a4z6 - 2a4z8 - 4a5z-1 + 16a5z - 24a5z3 + 18a5z5 - 6a5z7 + 4a6 - 13a6z2 + 11a6z4 - 5a6z6 - a7z-1 + 4a7z - 5a7z3 - a7z5 + a8 - 3a8z2

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                 1 j = 2               2   j = 0             3 1   j = -2           4 2     j = -4         4 4       j = -6       4 3         j = -8     2 4           j = -10   3 4             j = -12   3               j = -14 2

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 30]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 30]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 10, 4, 11], X[15, 5, 16, 20], X[7, 17, 8, 16], > X[17, 12, 18, 13], X[9, 14, 10, 15], X[13, 18, 14, 19], X[19, 9, 20, 8], > X[2, 5, 3, 6], X[11, 4, 12, 1]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, -2, 10}, {9, -1, -4, 8, -6, 2, -10, 5, -7, 6, -3, 4, -5, 7, > -8, 3}]
In[6]:=	Jones[L][q]
Out[6]=	2 5 6 8 7 7 5 3/2 ----- - ----- + ---- - ---- + ---- - ---- + ------- - 3 Sqrt[q] + q 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 -18 -16 2 5 -10 4 -6 2 4 -1 - q - --- + q + q + --- + --- + q + -- - q + q - q 20 14 12 8 q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 30]][a, z]
Out[8]=	3 5 7 a 4 a 4 a a 3 5 7 3 3 3 - - ---- + ---- - -- + 2 a z - 9 a z + 6 a z - a z + 3 a z - 9 a z + z z z z 5 3 5 3 5 5 5 3 7 > 4 a z + a z - 5 a z + a z - a z
In[9]:=	Kauffman[Link[10, NonAlternating, 30]][a, z]
Out[9]=	3 5 7 2 4 6 8 a 4 a 4 a a 3 1 + 4 a + 7 a + 4 a + a - - - ---- - ---- - -- + 3 a z + 15 a z + z z z z 5 7 2 2 2 4 2 6 2 8 2 > 16 a z + 4 a z - 3 z - 14 a z - 21 a z - 13 a z - 3 a z - 3 3 3 5 3 7 3 4 2 4 4 4 > 9 a z - 28 a z - 24 a z - 5 a z + 3 z + 13 a z + 21 a z + 6 4 5 3 5 5 5 7 5 6 2 6 4 6 > 11 a z + 10 a z + 29 a z + 18 a z - a z - z + a z - 3 a z - 6 6 7 3 7 5 7 2 8 4 8 > 5 a z - 3 a z - 9 a z - 6 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 4 2 3 3 4 2 4 4 3 -- + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 4 2 14 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q q q t q t q t q t q t q t q t q t 4 2 t 2 2 2 4 3 > ---- + 3 t + --- + t + 2 q t + q t 4 2 q t q

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